Inventory-based versus Prior-based Options Trading Agents

Transcription

1 Inventory-based versus Prior-based Options Trading Agents Abraham Othman and Tuomas Sandholm Computer Science Department Carnegie Mellon University January 27, 2012 Abstract Options are a basic, widely-traded form of financial derivative that offer payouts based on the future price of an underlying asset. The finance literature gives us option-trading algorithms that take into consideration information about how prices move over time but do not explicitly involve the trades the agent made in the past. In contrast, the prediction market literature gives us automated market-making agents (like the popular LMSR) that are event-independent and price trades based only on the inventories the agent holds. We simulate the performance of five trading agents inspired by these literatures on a large database of recent historical option prices. We find that a combination of the two approaches produced the best results in our experiments: a trading agent that keeps track of previously-made trades combined with a good prior distribution on how prices move over time. The experimental success of this synthesized trader has implications for agent design in both financial and prediction markets. 1 Introduction Many prediction markets today rely on automated market makers to price contracts. These are electronic agents that stand ready to buy and sell con- To appear in Algorithmic Finance. 1

2 tracts with participants and adjust the prices they offer in response to those trades. Prediction markets use these agents because there may not be enough organic liquidity in a market to have narrow bid/ask spreads, or the event space the prediction market runs over is so large that buyers and sellers could have trouble matching their orders without the help of an intermediary [Pennock and Sami, 2007, Chen and Pennock, 2010]. These automated market making agents typically price contracts solely based on the trades the marketmaking agent made in the past (i.e., the agent s inventory). Consequently, automated market makers are usually outcome-agnostic, which means that they can be applied to virtually any elicitation problem without particular domain knowledge. This unlocks the long tail of elicitation problems that are of specific, rather than general, interest. For instance, Inkling Markets, Crowdcast, and Consensus Point, which are start-up companies, offer clients the ability to use an internal prediction market mediated by an automated market maker to forecast the outcome of uncertain events that are of interest to only a small number of people (for instance, whether a project will be completed on time). These market makers are almost always used in fake-money applications, not with actual money on the line. In a prediction market, the goal of a market maker is often to assist in information elicitation, and so the losses that inevitably result from these market makers are viewed as subsidies to encourage traders to participate and reveal their information [Hanson, 2003, Pennock and Sami, 2007, Chen and Pennock, 2010]. This reasoning provides a contrast between a market maker in a prediction market, which has a designated role to provide liquidity and can lose money acceptably, and a trading agent in a financial market which speculates for its own account. We specifically study the latter in this paper. A trading agent will make or lose money based on whether its distribution over futures states of the world is better or worse than that of its counterparties, and an outcomeagnostic agent, the norm for making prediction markets, generally has too much entropy in its prior over outcomes to profit. The finance literature, and particularly the finance literature as it pertains to derivatives, has taken a different modeling approach than the prediction market literature. The modern derivatives literature started with the seminal work of Black-Scholes and Merton (BSM) [Black and Scholes, 1973, Merton, 1973]. By providing a practical way to price options contracts effectively, BSM led to the options markets that are the precursors of modern derivatives trading. Additionally, because the formula for calculating prices is entirely 2

3 self-contained, it provided the constructive groundwork for the notion of autarky within finance theory. The BSM formula takes only three inputs the current price, volatility, and the risk-free rate of return to produce an ensemble of options prices. Although it is problematic to generalize over the entire finance literature, speaking broadly, most models of asset pricing operate using autarky as a guiding principle. In an autarky model, prices are philosophically prior to the agents that trade on them, so these models have no reliance on an agent s past actions in determining prices. In this paper, we synthesize ideas from the prediction markets and finance literatures to create more successful trading agents than either literature on its own. The key insight in this paper is that these two notions: having good priors, and learning from inventories, are not oppositional. We combine them to create a trading agent that develops actionable prices based on both factors. Interestingly, the combination of the two ideas is not straightforward, and there are significant theoretical hurdles that serve to restrict the valid combinations of prior distributions and the utility models that determine how an agent reacts to its inventory. The principal contribution of our work is the experimental simulation of five different trading strategies on a large body of recent options data: 1. A zero-intelligence agent, added as an experimental control, that trades randomly. 2. An orthodox BSM Log-normal distribution trader. 3. A Normal distribution trader, with mean and variance matched to the Log-normal trader. 4. The Logarithmic Market Scoring Rule (LMSR), the most popular automated market maker in Internet prediction markets [Hanson, 2003, 2007, Pennock and Sami, 2007] which is equivalent to an exponential utility trading agent with constant uniform priors. 5. A hybrid agent that combines exponential utility with normal distribution priors. We find that by many different measures, including expected return and worst-observed performance, the hybrid trader outperforms the other traders. Consequently, our results support the hypothesis that a trader s current exposure can be a profitable influence on future actions, and that a trader can learn from their past actions to create a more accurate estimate of the future. 3

4 Even though the hybrid trader performs the best on several key metrics, it does not stochastically dominate the performance of the parametric traders from the finance literature. This means it is possible to construct a coherent utility function that would prefer the performance of the parametric traders. However, the relative performance of the hybrid trader relative to the parametric traders provides insights into the larger qualitative question of how the hybrid trader is able to perform well. We believe our results are best interpreted by the hybrid trader profitably insuring against the risk that its model of the future is inaccurate, a claim which we justify in detail. It is important to clarify what we believe is significant about our results. We do not believe that our hybrid trader is the best options trading agent that could be devised. Certainly, BSM can be considered a theoretical model, rather than a practical trading agent, and there are likely other agents that could be constructed that would have better performance on our dataset. These agents could employ more sophisticated models about how prices move through time, detailed order book information, or outside information like press releases and macroeconomic forecasts. But what is significant about our work is this: by incorporating inventories into a Normal distribution trader, we increase performance without incorporating any new or more sophisticated information into the trading process. Instead, we achieve these performance gains simply by paying attention to information that autarky models discard. As artificial intelligence researchers we are particularly keen on interpreting our results within the intellectual framework of our discipline. Much of modern, optimization-based AI is grounded in the idea of modeling (or actually constructing) a robot that can observe its environment, effect an action, and receive a reward that depends on its state and the actions it has chosen [see Russell and Norvig, 2003, Chapter 2]. This model fits naturally into trading options, where the environment is the market, the actions are the contracts to trade, and the rewards are literal monetary profits and losses. History-keeping is vital to what constitutes a rational agent in the AI literature. In a general environment an agent that does not retain its history cannot be rational. So in this context, our results are not surprising: agent performance is enhanced by both built-in knowledge about the future and the ability to learn from one s past actions. 4

5 2 Options In this section we introduce options and their associated terminology, as well as our dataset. 2.1 What are options? Options contracts involve the future opportunity to buy or sell some kind of underlying instrument (the underlying) at a set price (the strike price). There are two types of option contracts. A call option gives its holder the right to buy an obligation at a specified price, and a put option gives its holder the right to sell an obligation at a specified price. These contracts have a hinged form of payouts. Definition 1. Let the underlying expire at price π. A (European) call option with strike price s has value max(π s, 0). A (European) put option with strike price s has value max(s π, 0). Options that strike close to the current price of the underlying are known as at the money. Options that are valuable at the current price of the underlying (high-strike puts or low-strike calls) are known as in the money. Options that are worthless at the current price of the underlying (low-strike puts or high-strike calls) are known as out of the money. There are two principal ways that govern the exercise of options. European options expire in cash at a certain date, the strike date. In contrast, American options can be exercised for delivery of the underlying at any point before the strike date. The additional optionality of American options makes them necessarily at least as expensive as European options. However, this additional optionality is rarely exercised, making American options essentially European in practice. Varian [1987] provides a theoretical argument based on no-arbitrage principles for European and American options having exactly the same prices. We examine both European and American options in our dataset. Options with the same underlying and strike date form what is called an options chain. Chains consist of an ensemble of contracts along with their associated prices. For instance, a chain might consist of calls and puts with strikes of 900, 1000, and 1100 for the ^SPX underlying expiring on December 22, Each of these contracts has a price at which they can be bought or sold (the ask and bid prices, respectively), and theoretically all of these 5

6 prices are based on some underlying distribution over the expiration price. This distribution changes over time as the price of the underlying and the time until expiration changes. When we perform our experiments, we step through simulating trading agents on snapshots of each options chain as it evolves over time, from initiation until expiration. 2.2 Historical dataset The dataset we use covers almost seven years of data on eleven underlyings, taken at 15-minute intervals. It is comprised of nearly 300 million {underlying, expiration date, strike price, datetime, best bid, best ask} tuples, and the corresponding {underlying, datetime, underlying best bid, underlying best ask} tuples for the underlying. The data spans from January 2004 through September To our knowledge this is the one of the moredetailed datasets used in an academic study on options studies generally use data from daily closing prices, which is much less detailed (and one of the most widely-cited papers on empirical option pricing, Dumas et al. [1998], uses weekly data). Table 1 gives an overview of the dataset. In order to provide a clean train/test separation, we divide the first two years (January 2004 through December 2005) to learn the relevant parameters for our simulation, and only test on the remaining data. A naïve split between training and testing data that randomly placed chains or days into different partitions would be tainted, because the training and test sets would overlap temporally. Consequently, only options chains that expire in 2006 and beyond are in our testing set. The number of complete chains in the testing set for each underlying is noted in the Testing chains column of Table 1. 3 Agents based on Log-Normal and Normal Distributions In this section we introduce the first two traders we used in our experiments, the Log-normal and Normal distribution traders. These traders are derived from existing work in the finance literature. 6

7 Underlying Description Clearing Number of records Testing chains ˆSPX S&P 500 Index European 53.6 million 6 ˆDJX Dow-Jones Index European 43.7 million 9 ˆNDX NASDAQ Index European 30.1 million 3 ˆIRX Thirteen-week treasury bill European 3.4 million 19 ˆFVX Five-year treasury yield European 6.2 million 19 ˆTNX Ten-year treasury yield European 6.9 million 16 ˆXAU Gold-Dollar Index European 10.1 million 16 X US Steel American 9.3 million 5 C Citigroup American 7.5 million 6 GE General Electric American 7.7 million 5 MSFT Microsoft American 8.5 million 5 XOM Exxon-Mobil American 7.7 million 5 Table 1: Our options dataset includes a diverse mix of underlyings, including indices, bonds, commodities, and equities of very liquid to moderately liquid contracts cleared both in American and European ways. 7

8 3.1 The BSM model Option pricing was revolutionized and popularized by the work of Black and Scholes [1973] and Merton [1973] (BSM). Those authors described a parametric framework under which prices on the underlying change according to a log-normal distribution, which was the solution to a differential equation. This framework was essentially unchallenged until Black Monday of 1987, where stock prices dropped precipitously in a single day. After this, options now show a persistent volatility smile or volatility skew, where out-of-the-money options are overpriced relative to BSM [MacKenzie, 2006]. An interpretation of this phenomenon is that the log-normal distribution of future prices as predicted by BSM is inaccurate, and the skew represents an effort to make the predicted distribution heavier-tailed. Another interpretation is that investors use options to provide insurance against the state of the world in which extremely low values of the underlying are realized. 3.2 Calculating contract values when the underlying is log-normally distributed We use a constant (daily) volatility parameter σ for each underlying. These values are learned from our training data in order to assure a clean train/test separation. Table 2 shows the values we used in our experiments. In addition to the current price and the volatility parameter σ, the BSM model takes an additional input, the so-called risk-free rate of return. This value reflects the time-cost of money. In our exploratory data analysis over our training data we did not see significant changes in performance for different realistic values of the risk-free rate (between zero and five percent annualized). For consistency, in our tests we set this value equal to zero for all the trading agents. In practice, banks, market makers, and large hedge funds will have small risk-free rates over short time horizons. Furthermore, setting this rate equal to zero means that any returns generated are exclusively from options trading, and not from interest on passive income. Because of the popularity of the BSM model, the formulas to calculate prices from a log-normal distribution are well-known. The so-called partial expectation of the log-normal distribution has an analytic expression. Let f denote the density function and F the distribution function of a log-normal distribution parametrized by µ and σ. Then the partial expectation formula is: 8

9 Underlying σ ˆSPX ˆDJX ˆNDX ˆIRX ˆFVX ˆTNX ˆXAU X C GE MSFT XOM Table 2: The (daily) volatility parameters σ are learned by taking the MLE of the daily changes of each underlying in the training set. s ( ) µ + σ log s xf(x) dx = e µ+σ2 /2 Φ σ where Φ( ) is the distribution function of the standard normal distribution. When this value is known, the price of a call option can be calculated, because the price of a call option with strike s is s (x s)f(x) dx = xf(x) dx sf(x) dx s s ( ) µ + σ log s = e µ+σ2 /2 Φ s(1 F (s)). σ By the use of the well-known put-call parity we can calculate the price of a put option at the same strike. The parity says that the price of a call at a strike, plus that strike, equals the price of a put at that strike, plus the value of the underlying. (Throughout this work, when we buy or sell underlyings it is assumed that the position will be closed when the options expire.) Once we have calculated the value of the call, the only unknown value in the put-call parity equation is the value of the put. 9

10 3.3 Calculating contract values when the underlying is normally distributed We also implemented a trader that models the underlying s expiration price as a normal distribution, instead of a log-normal. This trader sets the mean and variance of the normal distribution to match that of the log-normal distribution. Normal distributions are a feature of much of the literature on market making in both prediction markets and finance. As O Hara [1995] discusses, theoretical frameworks often model underlying prices as normal distributions because the conjugate prior to a normal distribution (with known variance) is another normal distribution. Consequently, it is common for agents to have a normal prior distribution and then update that distribution to a posterior normal distribution as more (normally-distributed) information arrives. This allows agents in models to act rationally while still retaining closed-form expressions for analytical tractability. Examples of models using normal distributions to project the future price of an asset include the classic models of Glosten and Milgrom [1985] and Kyle [1985], as well as more recent models such as that of Das and Magdon-Ismail [2009]. In the context of options, however, normal distributions are conceptually much more problematic than log-normal distributions. This is because they have support over the whole real line, rather than just over positive values. Since negative values cannot exist as termination prices (shareholders are not personally liable for the debts of a company), this makes the normal distribution inherently unrealistic a. Recognition of this problem dates back to some of the earliest work on asset pricing [Merton, 1971]. Reflecting this intuition, our results showed that the normal distribution trader generally performed worse than the log-normal distribution trader, even though the two traders matched the mean and variance of their distributions. To solve for prices using a normal prior, we used Gauss-Hermite quadrature [Judd, 1998]. Let E µ,σ (f) denote the expectation of the function f under a normal distribution with mean µ and standard deviation σ. Gauss-Hermite a One way to overcome this limitation is to instead deal with truncated normal distributions, where the probability mass that exists below zero is re-distributed over the positive values (e.g., Chang and Shanker [1986]). However, these concerns are more theoretical than practical, because in our simulations the total probability mass for negative realization values appeared to be small enough that truncating the distribution would not have changed agent actions. 10

11 quadrature provides a set of nodes x i and weights w i such that E µ,σ (f) i w i f(x i ) by implicitly converting the function f to an approximation by orthonormal polynomials. 4 The LMSR as an option trading agent In this section we discuss how we implemented the LMSR, the de facto automated market maker used in Internet prediction markets, as an options trading agent. 4.1 Background Internet prediction markets typically do not have enough organic liquidity or interest to support trade. Therefore, prediction markets can benefit from automated market makers algorithmic agents that provide liquidity. In a typical market of this type, traders place bets directly with the automated market maker, who then adjusts the prices of the bets offered to traders in response to the inventory the market maker now holds. The presence of an automated market maker gives even the least (organically) liquid markets a relatively tight bid/ask spread to facilitate price discovery and the ability for traders to liquidate their positions whenever they choose. Introductions to automated market making can be found in Pennock and Sami [2007] and Chen and Pennock [2010]. Unlike market makers in financial markets, the automated market makers of prediction markets are designed to facilitate trade first and make profits second. In fact, many prediction markets are designed so that the market maker will lose money in expectation to the set of traders; these losses are viewed as a subsidy paid from the market administrator to the traders to pay for the information they supply. Because of this subsidy, most Internet prediction markets use artificial currencies or raffle tickets, rather than real cash. The most popular automated market maker used in Internet prediction markets is the LMSR, an automated market maker with particularly desirable 11

12 properties [Hanson, 2003, 2007]. The LMSR is used by a number of companies including Inkling Markets, Consensus Point, Yahoo!, Microsoft, and the large-scale non-commercial Gates Hillman Prediction Market at Carnegie Mellon [Othman et al., 2010]. The LMSR is also the focus of academic studies about market microstructure [Othman and Sandholm, 2010b] and laboratory studies of market maker performance [Das, 2008, Brahma et al., 2010, Chakraborty et al., 2011]. We proceed to give a mathematical formulation of the LMSR. 4.2 Representation Let the space of possible futures be exhaustively partitioned into n events, {ω 1,..., ω n }, so that exactly one ω i will be realized. Let x be a vector of payouts, so that x i is the amount the market maker must pay out if ω i is realized. The LMSR is a cost function that translates these payout vectors into a single scalar value. The LMSR is given by ( ) C(x) = b log exp(x i /b) Where b > 0 is an exogenous constant known as the liquidity parameter. Larger values of b correspond to larger worst-case losses by the LMSR, which loses at most b log n. On the other hand, larger values of b produce tighter bid/ask spreads. When b = 1, the LMSR is equivalent to the entropic risk measure from the finance literature [Föllmer and Schied, 2002], however, the techniques were developed independently. (Agrawal et al. [2009] discusses the similarities between automated market makers and risk measures.) Automated market makers work by charging a trader that wishes to move the market maker s payout vector from x to y the amount C(y) C(x). Consider, for instance, an automated market maker pricing bets on a baseball game between the Red Sox and Yankees, so that the events are ω 1 = Red Sox win and ω 2 = Yankees win. Assume that b = 100, and that the market maker s current state is (300, 200), so that the market maker must pay out 300 dollars if the Red Sox win and 200 dollars if the Yankees win. Now, imagine a trader wishes to buy a bet that would pay out 50 dollars if the Red Sox win. Accepting that bet would change the market maker s payout vector from (300, 200) to (350, 200). Consequently, the market maker i 12

13 quotes the trader a price of C((350, 200)) C((300, 200)) 39 dollars for the bet. One feature of the LMSR is that the gradient of its cost function produces a probability distribution over the states i C(x) = exp(x i/b) j exp(x j/b). 4.3 Compressing the state space For the log-normal and normal distributions, we took the view that the underlying would expire as a continuous process. In reality, the space of expirations is countably infinite, delimited by one-cent intervals. It is natural, however, to reduce this infinite space to a finite range of possibilities. This is a lossy operation; there is perhaps the chance the final outcome will fall outside the range we specify. Therefore, we should take a wide range. Consider the ^SPX underlying, which tracks the S&P 500; it has a value of around It is reasonable to assume that for near-term options, its expiration price will be between 100 and 10,000. With one-cent discretization, this implies a space of about n = 1 million events. Very large event spaces like this pose two problems for the LMSR; one practical, and the other theoretical. First, the LMSR is numerically unstable over large event spaces; this problem was noted in the Gates Hillman Prediction Market [Othman and Sandholm, 2010a], and was also a problem in Yahoo s Predictalot b (which ran over a combinatorially large event space). This numerical instability makes implementing the LMSR over very large event spaces challenging, unwieldy, and potentially inaccurate. The second problem with large event spaces is that they correspond to larger worst-case losses; in order to maintain realistic worst-case losses the b parameter would need to be much smaller, leading to a large bid/ask spread that would not facilitate much trade. In full form, the size of the event space we would need to consider makes applying the LMSR to options markets extremely challenging. Fortunately, we can achieve a significant lossless dimensionality reduction that makes b 13

14 automated market making for options feasible. The key to compressing the state space is to focus only on the strike prices, not the expiration prices. Let the set of ordered strike prices be given by s = {s 1, s 2,..., s n } and the market maker have corresponding payout vector x = {x 1, x 2,..., x n }. This idea, of compressing the state space only to relevant strike prices, is a feature of many options trading models (e.g., Varian [1987]). Lemma 1. Let the expiration price be s i < s < s i+1, such that s = αs i +(1 α)s i+1. Then if there exist no contracts written for a strike price between s i and s i+1, the market maker must pay out αx i + (1 α)x i+1. Proof. An options contract is piecewise linear with a joint at its strike price, underlyings are linear, and our portfolio consists only of these contracts. Since a combination of linear functions is also linear, if no contract has a strike price between s i and s i+1, our payoffs will move linearly with the realized price between s i and s i+1. This linearity result allows us to, in effect, collapse the continuous state space of possible prices [s 1, s n ] into the set of discrete prices {s 1,..., s n } by bounding our realized loss. Proposition 1. Let x = max i x i represent the maximum value the market maker must pay out if s {s 1,..., s n } is realized. Then x is also the maximum value the market maker must pay out if s [s 1, s n ] is realized. If a value outside of the strike price range is realized, we might lose more than what would be suggested by the x i. One way to solve this problem is to include two dummy strike prices: one at zero, and another at an arbitrarily large value. This essentially prevents the final state from falling outside of the span of the strike prices. While this is appropriate for more general settings (e.g., sparsely-traded underlyings) we did not implement these dummy strikes into our LMSR trading agent. We found that for our underlyings the extreme strike prices of our option chains generally gave reasonable bounds on the expiration price. 4.4 Implementation details Recall that trades are priced in the LMSR based on the vector of payouts currently held by the market maker. In this section we describe how to go from an inventory of options contracts to a payout vector. Let {s 1,..., s n } 14

15 be the ordered set of strikes we are considering, and x = {x 1,..., x n } be the payout vector corresponding to the realization of each strike. Formally, selling a call at strike s corresponds to the payout vector x = {x i } n i=1 where { si s if s x i = i s 0 if s i < s Selling a put at strike s corresponds to the payout vector { 0 if si > s x i = s s i if s i s Selling the underlying corresponds to a payout vector of x i = s i The payout associated with an event (i.e., x i ) depends directly on the strike price associated with that event (i.e., s i ). For example, selling a contract of the underlying corresponds to a payout of 30 dollars if the underlying expires at 30, and a payout of 50 dollars if the underlying expires at 50. Selling a call at a strike of 20 corresponds to a payout of 0 if the underlying expires at 20 (or below), but a payout of 30 if the underlying expires at 50. Buying any contract induces the negative payout vector of selling that contract. Observe that when we quote the price to buy a contract the value will be negative, suggesting that we need to compensate our counterparty (i.e., pay out money for) the contract in question. The set of strike prices we model for the LMSR trader is all the currentlyoffered strike prices, plus any strike prices corresponding to contracts we traded in the past. Given an inventory I of bought and sold contracts, we can calculate the payout p i at any strike s i by doing an element-wise sum for the payout vector x of each accumulated contract: p i = x I x i This cumulative payout vector p is then used to price the available options in the chain; if a prospective contract induces a payout vector y the LMSR trader prices the contract at C(p + y) C(p) 15

16 The final issue is how to set the liquidity parameter b. For our experiments we set b equal to 2500 times the initial underlying price, a value that yielded good performance over the training data. Values much smaller than this resulted in sharply diminished trade because the bid/ask spread was too large. Values much larger than this resulted in marginal prices which stayed close to a uniform distribution for the entire trading period. 4.5 The LMSR as a constant-utility cost function In this section we introduce an alternate formulation of the LMSR as a constant-utility cost function [Chen and Pennock, 2007]. The cost functions define a utility function and then charge interacting traders precisely as much as required to maintain constant expected utility. In the finance literature, these schemes are known as indifference pricing [Carmona, 2009]. Definition 2. Let u : R R be an increasing concave function, π be a probability mass function over the ω i, and x 0 dom u. A constant-utility cost function implicitly solves π i u(c(x) x i ) = u(x 0 ) i The marginal prices of a constant-utility cost function are given by i C(x) = π iu (C(x) x i ) j π ju (C(x) x j ) (see Jackwerth [2000], Chen and Pennock [2007] for details.) By recalling that the marginal prices in the LMSR are given by i C(x) = exp(x i/b) j exp(x j/b) it is easy to see that the LMSR is equivalent to a constant-utility cost function with π i = 1/n and u(x) = exp( x/b). In words, the LMSR is an exponential utility agent with a uniform prior over the events. In the next section, we explore how to change the prior distribution from uniform to something more accurate. 16

17 5 Trading agents based on constant exponential utility We have presented two different ways of thinking about how to price options. The first is the traditional approach from finance, derived from projecting a distribution over the future and pricing obligations based on this projection. The second approach is derived from automated market making in prediction markets. It involves pricing obligations based only on trades previously made. In this section, we explore how to achieve a synthesis between these two ideas, creating a market maker with a good prior that also responds to past trades. This effort is immediately complicated by the fact that models from finance typically involve generating continuous distributions over the final strike price, while the LMSR, as we discussed in the previous section, is for discrete distributions. In order to synthesize these two models one needs to either discretize a continuous distribution, or develop a continuous analogue to the LMSR. Since the existing literature in financial options is based around continuous distributions, we choose to do the latter in order to better align our work with that literature. (We study a version of the LMSR with a good discrete prior in Appendix C. We found that it performs better than the maximum-entropy LMSR, but much worse than the trader developed in this section.) We synthesize the two methodologies by developing a version of the LMSR over continuous spaces by viewing it as a constant-utility cost function.we can generalize the constant-utility cost function framework we described in Section 4.5 to continuous event spaces. In the continuous setting, our payout vectors are functions x : R R, and cost functions become functionals that map these functions to scalars, C : (R R) R. Definition 3. Let µ be a probability distribution over possible expiration prices, u : R R be an increasing concave function, and x 0 dom u. A continuous constant-utility cost function C(x) is given implicitly by the solution to 0 µ(t)u(c(x(t)) x(t)) dt = u(x 0 ) Given this framework, it seems like it would be straightforward to combine the LMSR with the orthodox BSM forecasting model: simply set µ to be equal to the appropriate log-normal and set u equal to exponential utility. 17

18 Surprisingly, this approach does not work as planned and instead produces undefined prices for simple actions. 5.1 The undefined prices phenomenon In a nutshell, what produces undefined prices is the tradeoff between how quickly the tails of the agent s prior distribution fall off, and how severely the trading agent s risk aversion reacts to extreme losses. If the aversion to large losses is strong enough, it can outweigh the very small probabilities associated with those large losses. The resulting trading agent would not offer to trade a contract that could produce those losses at any price. The specific failure of exponential utility and log-normal priors to produce always-defined prices is known in the finance literature [Henderson, 2002], but working through a realistic example will shed light on how and why this pairing fails. We will then generalize the intuition gained from the example to multiple trades, distributions, and utilities, with a particular focus on what happens with exponential utility Example For this example, we will assume a log-normal prior with µ = 5 and σ = 0.5. Now consider the calculation involved in pricing the sale of the underlying. The sale of the underlying is given by the payout vector (function) x(t) = t. With exponential utility, recall that the cost function solves, for some v < 0: 0 e C(x) x(t) ( b ) µ(t) dt = v Because of the form of the utility function, we can uncouple the cost, which does not feature the dummy integrating variable t, from the vector of payouts 0 e x(t)/b µ(t) dt = ve C(x)/b which shows that the cost function is defined if and only if 0 e x(t)/b µ(t) dt converges. For our specific example, with the sale of the underlying and the log-normal distribution, this integral is 18

19 Weighted utility Expiration price Figure 1: Because exponential utility increases faster than log-normal probability falls off, our sensitivity to large losses grows unboundedly. 0 e t/b ( e (log t µ)2 /2σ 2 t 2πσ 2 Figure 1 shows a plot of the integrand. The x-axis is log-scaled. The integrand tends towards zero for large expiration values (in the thousands, the median of the prior distribution is e 5 148). However, for unrealistically extreme expiration values (more than 100 times the fictional current price) the integrand explodes negatively, and the integral diverges. Why does this occur? It is because for extremely large realization values our sensitivity to the prospect of extreme loss (since we are selling the underlying we lose when it expires high) outweighs the extremely small probabilities that the log-normal distribution produces for those values. We can show that this behavior holds regardless of the particular b, µ, σ parameterization chosen. Again, consider the pricing integral corresponding to the sale of the underlying. With a log-normal distribution and exponential utility, we have µ(t) Θ(e log2 t ) and u(x(t)) Θ(e t ), so ) u(x(t))µ(t) Θ(e t log2 t ). dt 19

20 Since lim t t et log2 0, the pricing integral diverges. It is easy to see that this asymptotic analysis also applies to selling a call, since when we sell a call with strike price s, u(x(t)) Θ(e t s ) Θ(e t ). Consequently, there exists no amount of money an exponential utility cost function with a log-normal prior would sell an underlying or a call for The theoretical basis of undefined prices For any probability distribution, the density at extremely large realizations is very small, but our utility function could react to these realizations in a pronounced way. Hence, we have a tension between the density of the distribution at its tails and the response of the utility function. In this section, we provide theoretical bounds for this phenomenon, motivated by demonstrating why using exponential utility with normal priors succeeds, while with log-normal (and many other) priors it fails. Geweke [2001] also explores diverging (undefined) prices, given certain priors and utilities, but only for a specific family of utility functions. The following result rules out using utility functions like log and 1/x with infinite-domain probability distributions. Proposition 2. In a continuous constant-utility cost function, if the prior probability distribution is defined over (0, ) but the utility function is not defined over all of R, then there exists a transaction with undefined price. Proof. We will show that the utility function will be evaluated at an undefined value. Let x(t) = t denote the function corresponding to selling an underlying. By assumption, the utility function is not defined at v R. Now, consider C(0) > v. Because the utility function is increasing, the cost function is increasing, and therefore C(x) > C(0). But because C(x) is finite and the prior distribution is defined at every t (0, ), there exists some t for which C(x) x(t) = C(x) t = v, so the utility function will be evaluated at v, producing an undefined value. This argument holds by symmetry in the case where C(0) < v, in which case we would buy the underlying instead to force the undefined evaluation. 20

21 The following result shows that one way to achieve well-defined costs is to only consider finite probability distributions. Proposition 3. In a continuous constant-utility cost function, if the utility function is defined and finite over all of R and the prior distribution is bounded, so that there exists a T such that for all t > T, µ(t) = 0, then the cost function is well-defined. Proof. The limits of integration are finite and the utility function is defined and finite for any argument. It follows that the integral to determine the cost function is always well-defined. One application of this result to the option trading problem is to consider a trading agent with a log-normal prior distribution that is truncated above some upper boundary T. (The remaining probability mass above the upper bound could be re-distributed below the bound, or, because it is likely to be so small in magnitude, safely ignored.) Proposition 3 suggests then that the cost function would always be defined when using such a distribution. However, this truncation scheme is numerically hazardous because it provides no guidance for which upper bound of T to select. Consider again Figure 1, which shows the weighted utility of a contract. Since the utility of the contract calculated without an upper bound on the prior diverges, where the upper bound T is set will have a great effect on the calculated utility of taking on the contract. To be concrete, it is clear from inspection that setting the upper bound at 1,000, 10,000, or 100,000 will produce vastly different calculations for the utility of the trader, and therefore for the fair price of the contract. Now specifically focusing on exponential utility, it turns out we must have very light tails for the cost function to be well-defined. Proposition 4. A continuous exponential-utility cost function is defined for every set of options transactions if and only if is bounded for every c 0. 0 e c t µ(t) dt Proof. Recall we have defined prices if and only if 0 e x(t) b µ(t) dt 21

22 is well-defined. By removing constant factors, we see that 0 e x(t) µ(t) dt must be well-defined. Options contracts are continuous, piecewise-linear functions, so therefore x(t) is a continuous, piecewise-linear function. Therefore there exists some c such that x(t) c t, where the bound is tight by selling c underlyings. Now since e x is an increasing function, this implies 0 e x(t) µ(t) dt 0 e c t µ(t) dt so if the right-hand equation is finite, the left-hand equation is finite also. Recall that the expression 0 e c t µ(t) dt is also known as the moment generating function of the distribution µ. However, moment generating functions are generally used only for the values of their derivatives at the argument c = 0, whereas for our result here the function must be defined for any positive c. This result has the effect of significantly limiting what distributions we can use with exponential utility. We have already discussed how we cannot use the log-normal distribution, but other distributions, like the chi-square, exponential, and Weibull are also ruled out. (This begs the question of whether there is a theoretical or empirical reason to use these distributions.) Now, consider that for the normal distribution, µ(t) Θ(e t2 ) This means that for arbitrary c the integrand is Θ(e c t t2 ) which results in a well-defined integral for any c. This means we can combine a normal distribution prior with exponential utility and still get defined prices. 22

23 5.2 Our Exponential utility trader For our Exponential utility trading agent, we use the same normal distribution prior as the Normal trader and the same b parameter as the LMSR trader. This is to better facilitate comparisons between the traders. 6 Random agent as a control In order to provide a performance benchmark, we introduce a random trading agent as a control in our experiments. Random is a trading agent that performs a uniform random action over the set of possible actions at each time step. That is, for all the bids and asks on the relevant calls, puts, and the affiliated underlying, Random selects a bid or ask to trade uniformly at random. Random can be thought of as a zero-intelligence agent [Gode and Sunder, 1993] that does not learn or optimize. We would expect Random to consistently produce slightly negative returns. Consider that the bid and ask prices are spread, so we should expect that an agent that takes either side of the market at random should consistently eat this spread. That is, we can roughly think of the Random trader as buying an option at price p + ɛ and then selling it at p ɛ, resulting in a guaranteed loss of 2ɛ. 7 Experimental setup As we discussed in Section 2, we simulated the performance of the trading agents on each options chain in our testing set. In our simulations, we step through 15-minute increments on each chain from initiation to expiration. Figure 2 shows a flowchart of the simulation steps over each testing chain (recall the number of testing chains for each underlying is listed in Table 1). Appendix A features a worked example of the simulation process with each trading agent on one snapshot of a single option chain. Any simulation on historical data is fraught with the risk of overfitting, producing an unreasonably rosy picture of real-world performance. We took several steps to combat overfitting. These limits are intended to give a more accurate picture of live performance than a naïve optimization over the dataset. 23

24 Agent observes market prices Agent computes fair prices for available contracts The set of favorable trading opportunities is determined Exactly one contract from the set, selected uniformly at random, is traded Time advances 15 minutes NO Does the chain expire now? YES Agent receives payoff from inventory Figure 2: The steps taken by an agent over each option chain. Agents differ in the second step, the way they determine the fair prices for the offered contracts. It is, of course, impossible in hindsight to accurately produce a counterfactual answer to the question of how well a trading bot would have performed. The bids and asks we fill could cause changes in the behavior of other traders, which is an effect we cannot judge from past data. However, Even-Dar et al. [2006] suggest that, as long as agents in the market trade on absolute values (fixed prices) rather than relative values (based on the current state of the order books), overall price series will be resistant to the small changes produced by simulation. Options markets are driven mainly by BSM-style models and often feature relatively large (compared to the underlying) bid/ask spreads. Therefore, we believe that these markets are populated mostly by traders of the absolute, rather than the relative type, making simulation by an agent trading a small amount of total volume appropriate. Another concern related to the difference between real trading and simulation involves the effect of adding orders. In the real world, we would be able to add our own limit orders to the order book so that we would both provide as well as consume liquidity. However, for robustness we do not model this property within our data; we only take prices and do not simulate the effect of adding limit orders to the dataset. Presumably, adding the ability to place limit orders at auspicious prices would only help the performance of these trading agents in the real world. 24

25 We take two more steps to handicap the performance of our trading agents to avoid overfitting to the historical data: We limit the frequency of trading to only a single contract every 15 minutes. We are concerned about slippage the tendency of prices to move against a trader s actions as the trader absorbs liquidity. Trading a single contract is a conservative measure of performance that avoids slippage, because when a counterparty sets a price, the counterparty must sell at least one contract at that price. We choose which contract to trade uniformly at random from the set of desirable contracts. Say that our algorithm has identified purchasing an underlying, selling a call at 20, and buying a put at 30 to be beneficial trades given the current market prices. Then we will do only one of these, each with probability one third. This is the case even if, say, our trading algorithm thinks that buying the underlying would be better than selling the call at 20. This restriction is designed so as to not overfit on our static snapshots of prices. In a real setting, trading opportunities may arise and disappear quickly and we may not be able to trade the best opportunity that exists at a given moment. We contrast the results of trading a random contract with trading only the contract an agent thinks is best in detail in Appendix B. Our results show the former model disadvantages the inventory-based traders more than the finance literature traders, so this restriction is conservative, as desired, for the conclusions we will draw in Section 8. Furthermore, when we examined the trading behavior that arose from only trading the best contracts, we saw that it was biased towards trading the same contracts again and again. This is unrealistic behavior, because presumably the liquidity associated with those contracts will be exhausted if they are traded so frequently, and the prices would slip. Since a trading agent generally has several contracts it is interested in trading at a given time step, trading uniformly at random produced more realistic behavior in the simulation by spreading trading activity among several contracts. 25

26 8 Results We begin by providing the numerical results from our experiments and then discussing those results qualitatively. 8.1 Quantitative results In terms of real-world trading performance, it would be most appropriate to quantify performance of trading agents in terms of net annualized return (e.g., 10% a year ). Of course, net return is a function of both value generated as well as value risked. Because of the form of the options contracts, determining the value risked is not straightforward. Single contracts, like selling a call, could lose an unbounded amount of money in the worst case. Combinations of options could amplify or hedge these losses. In practice, traders need to put up a certain fraction of their positions with the exchange (the margin) in order to maintain those positions. The precise margin amount depends on the rules of the particular exchange the options are traded on and is generally based around historical models of how prices move over time. While percent return is difficult to calculate and depends on a host of practical matters, net performance (gain or loss) is simple to calculate. Therefore, we use net performance as a measure of trading agent performance. To normalize net performance, we divide a trading agent s gain or loss for a chain by the initial underlying price and by the number of days the chain is active. The resulting figure gives a meaningful way to compare chains where the underlying is in the thousands (like ˆSPX) or the ones (like C), over differing numbers of days. We refer to this normalized value as net underlyings per day (NUPD). Table 3 provides summary statistics for the performance of each trading agent over the 114 testing chains in terms of NUPD. The Exponential utility agent had the most positive instances, highest mean, and best worst-observed performance. The Log-normal agent had the highest median performance. The Random trading agent had the worst performance along each of these dimensions. To assess the significance of the values in Table 3, we performed a bootstrap analysis that simulated running our experiments 10,000 times. Bootstrap analysis was a natural choice for this setting because of the complexity of estimating e.g., worst-observed loss using standard techniques [?]. The head-to-head results of this analysis is given in Table 4. For instance, the 26